## The Z mass at a hadron collider November 25, 2008

Posted by dorigo in personal, physics, science.
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The Z boson mass has been measured with exquisite precision in the nineties by the LEP experiments ALEPH, OPAL, DELPHI and L3, and by the SLD experiment at SLAC: we know its value to better than a few MeV precision. The PDG gives $M_Z = 91.1876 \pm 0.0021 GeV$. Now, a precise Z mass is an important input to our theory, the Standard Model, and through its measurement, as well as that of other Z-related quantities that the four LEP experiments and SLD measured with great precision, a giant leap forward has been made in the understanding of the subtleties of electroweak interactions.

For an experimental physicist, however, the knowledge of the Z mass is more a tool for calibration purposes than a key to theoretical investigations. Indeed, as I have discussed elsewhere recently, I am working at the calibration of the CMS tracker using the decays of Z bosons, as well as of lower-mass resonances. We take $Z \to \mu \mu$ decays, we measure muon tracks, determine the measured mass of the Z boson with them, and compare the latter to the world average. This provides us with precious information on the calibration of the momentum measurement of muon tracks.

In CMS we will quickly collect large numbers of Z bosons, so statistics is not an issue: we will be able to study the calibration of tracks very effectively with those events. However, when statistics is large, experimentalists start worrying about systematic uncertainties. Indeed, there are several effects that cause a difference between the mass value we reconstruct with muon tracks and the true value of the Z boson mass -the one so well determined which sits in the PDG.

I decided to study one of those effects today: the mass shift due to parton distribution functions (PDF). When you collide protons against other protons, what creates a Z boson is the hard interaction between a quark and an antiquark. These constituents of the projectiles carry a fraction of the total proton momentum, but this fraction -called parton distribution function– is unknown on an event-by event basis. By studying proton collisions in different conditions and environments for a long time, we have been able to extract functions $f_q(x)$ which describe how likely it is that a quark q in the proton carries a fraction x of the proton’s momentum. As an example, if the proton travels at 5 TeV as in LHC, an x value of 0.1 means that the quark q will carry 500 GeV by itself.

Now, things are complicated, because each different quark q (u,d,s,c,b) has its own different parton distribution function. The proton contains two valence up-quarks and one valence down-quark: it has a (uud) composition. Those quarks carry a good part of the proton’s momentum, but a large share is due to the rest of partons the proton is made of: sea quark-antiquark pairs, and gluons. Protons do carry antiquarks of all kinds -five in total-, as well as gluons, and these, too, get their own distribution function. A plot of the parton distribution functions of the proton (with a logarithmic x-axis to enhance the low-x behavior) is shown on the right. Note the bumps of u- and d- quark distributions, in blue and green, respectively: those bumps are due to the valence quark contributions.

In reality, things are even more complicated than what I discussed above: you do not simply get away with one function per each of the 11 partons I mentioned thsi far, because these functions have a value which depends on the energy at which you probe the proton, $Q^2$: in a soft collision (which means a small $Q_1$), $f(x, Q^2_1)$ is very different from what it is in a harder one, $f(x, Q^2_2)$ (with a larger $Q_2$).

The reason for the weird behavior of parton distribution functions -their evolution with $Q^2$– is that quarks have the tendency of emitting gluons, becoming less energetic, and this tendency in turn depends on the energy Q at which they are studied. What is stated above is encoded in very famous functions called DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) equations. They are in a sense another consequence of the “asymptotic freedom” exhibited by strongly interacting particles: at high energy they behave as free particles, emitting little color radiation, while at low energy their interaction with the gluon field increases in strength. It is all due to the fact that the coupling constant of the theory, $\alpha_s$, is large at small Q. That constant is not a constant by any means!

You have every reason to be confused now: I was talking about calibrating the CMS tracker using muons, and now we are deep into Quantum ChromoDynamics. What gives ? Well: Z bosons are created by quark-antiquark annihilations, and those are found inside the colliding protons with probabilities which depend on their momentum fraction x, and on the total collision energy Q. Since the PDF of quarks and antiquarks peak at very small values of x, the probability of a collision yielding a Z boson -which has a respectable mass of 91 GeV- is small.  If the Z was lighter, more of them would be produced. Now, the Z boson is a resonance, and like every resonance, it has a finite width. What that means is that not all Z bosons have exactly the same mass: while the peak is at 91.186 GeV, the width is 2.5 GeV, which means that it is not infrequent for a Z boson to have a mass of  89, or 92 GeV, rather than the average value. This is described by the Z lineshape, a function called Breit-Wigner:

$F(\Gamma,M) = \frac{\Gamma/2} {(M-M_Z)^2 + \Gamma^2 /4}$.

The function is shown below.

As you can see, there is a non-negligible probability that a Z boson has a mass quite different -even a few GeV off- from 91.19 GeV. Now, since Z bosons can be created at masses lower than $M_Z$, they will be privileged by parton distribution functions over masses higher than $M_Z$ by the same amount, because .parton distribution functions are larger at lower x. This creates a bias: the perfectly symmetric Breit-Wigner lineshape gets distorted by the preference of partons to carry a lower fraction of the proton momentum.

The distorsion is very small, but it is very important to take it in account when one wants to use measured Z masses to precisely calibrate the track momentum measurement. To size up the effect of the PDF on the Z lineshape, one can compute an integral of the Breit-Wigner weighted with the PDF $f(x)$, by taking into account the different combinations of quarks which give rise to a Z boson in proton-proton collisions.

A Z can be produced by the following quark-antiquark interactions:

• $u \bar u$: this can originate from a valence u-quark and a sea anti-u-quark, as well as from a sea u-quark and a sea anti-u-quark. The probability that this quark pair creates a Z depends on the coupling of u-quarks to the Z boson, and this probability is a function of some coefficient predicted by electroweak theory. It is proportional to 0.11784.
• $d \bar d$: same as above, but the coupling is proportional to 0.15188.
• $s \bar s$: these can only occur through sea-sea interactions. The coefficient is the same as for d-quarks.
• $c \bar c$: these are due to the small charm component of the proton sea. They get the 0.11784 coefficient as u-quarks too.
• $b \bar b$: these are tiny, but still exist. b-quarks couple to the Z with the 0.15188 factor.
• $t \bar t$: these are basically zero.
• $g g$: gluon-gluon collisions cannot produce a Z boson, because they are vector particles as the Z (spin 1), and a vector-vector-vector vertex is zero by construction. Note that the same does not hold for the Higgs boson, which is a scalar (spin 0) particle: a vector-vector-scalar vertex is possible, and in fact it is the largest contribution to H production at the LHC.

Putting everything together, one may compute the shift in the lineshape of the Z, and plot it directly (right, on a logarithmic scale to show the effect on the tails), or as a function of the rapidity of the Z boson, a quantity labeled by the letter Y (the dependence is shown in the last graph of this post, below). Rapidity is a measure of how fast is the Z boson moving in the detector reference frame: when one of the partons has a much larger momentum fraction than the one it is colliding against, the produced Z boson has a large momentum in the direction of the more energetic parton.

The rapidity distribution of Z bosons is shown in the graph below, separately for Zbosons produced by valence-sea collisions (in red) and by sea-sea collisions (in blue).

A rapidity Y=0 means that the Z was produced at rest in the detector, +5 is a fast-forward-moving Z, and -5 is a Z moving in the opposite direction with as much speed. As you can see, the valence-sea interactions are the most asymmetric ones, predominantly producing a forward-moving Z boson.

On the right here I also plot with the same color-coding the x distribution of quarks taking part in the Z creation. The red distribution has both a very small-x and a very large-x component, highlighting the asymmetric production.

Despite being in black and white, the most interesting plot is however the following one. It shows the average mass of the Z bosons (on the vertical scale, in GeV) as a function of the Z rapidity. The downward shift from 91.186 GeV is relevant -about 0.25 GeV overall- but it increases at large values of rapidity, when one of the two partons has a very small value of x, so that the collision “samples” a rapidly varying PDF for that parton.

The plot on the left here is what is needed as an input for our calibration program: we will have to study how this dependence affects our determination of the momentum scale. A lot of work ahead, but a very enlightening one!

1. pedro - November 25, 2008

Hi Tommaso,

Do you understand the deep on the Z mass around +/-4 ? It does rise again after it. It might be related with the peak in y=4 for Z rapidity in the red curve, but I failed to see how. As I understand it should not be related with cuts on your simulation input parameters, right?

Pedro.

2. dorigo - November 25, 2008

Hi Pedro,

the dips at Y=+-4 are physical, the points above that are just due to insufficient sampling of the PDF in the computation of the factorization integral. The dips are due to the fact that at very high rapidity the PDFs sample very small values of x, where the behavior is more acute than just 1/x-like. Thus, the Breit-Wigner gets a larger bias towards smaller values.

Cheers,
T.

3. dorigo - November 25, 2008

… to be clear, data at |Y|>4.2 is meaningless… But the trend below that value is real.

4. Stefan - November 25, 2008

Hi Tommaso,

thanks, very interesting post! BTW, beam rapidity Is about +/-9.6 for the LHC? So, the two bumps in the Z rapidity distribution are way within the total rapidity range? Is there an easy argument to see why they are at +/-4, and indeed, why this funny dip in the average mass at this rapidity?

Cheers, Stefan

5. Stefan - November 25, 2008

Sorry, comments crossed… so the dropping at +/-4 is real, but the raise at even larger |y|, creating the impression of a dip, is an artifact of insufficient statistics?

6. pedro - November 26, 2008

Hi Tommaso,

The trend for high rapidity to be biased towards smaller values on the BW I understand. I am not sure if I buy your explanation that |Y|>4.2 is meaningless. From the rapidity plot we see that |Y| = 4.5 has more than half the numbers of events as Y=0, so there is enough events to sample it. Maybe I miss your point.

Pedro.

7. Luboš Motl - November 26, 2008

Dear Tommaso,

a decrease of a particle rest mass as a function of its rapidity is “physical”? If you mean a similar thing by “physics” as I do, can I ask you whether you claim that you have just proved a violation of the Lorentz symmetry?

You have claimed many big – and three times bigger – discoveries recently, so my shock would be less intense than before. 😉 Do you agree that the independence of the rest Z mass on the speed or rapidity follows from special relativity and if you reconstruct this quantity accurately, it should also follow from your experiments?

Best wishes
Lubos

8. dorigo - November 26, 2008

Hi Stefan,

yes, I believe that the points at |Y|>4 to be affected by the way I do my integral,and to be meaningless. We must not get deceived by the symmetry of the data points, which is just coming from the parity properties of the integral.

Hi Pedro,

yes, there is enough stats in the bins, but those stats come from ill-sampled x1,x2 values. You do not get to see it from the Y distribution: you would only see it if I plotted the x values of the events in those tails. You would then get to see a bumpy distribution, because I am sampling x in steps of 10^-5 uniformly, in a region where the f(x) varies wildly. I can see the same behavior arising at Y=3.5 if I only sample every 10^-4, and at still lower Y if I do an even coarser sampling.

Hi Lubos,

the rest mass of the Z is what it is, while I am plotting the average mass of Z bosons produced in pp collisions, which is a different thing. The Breit-Wigner is always the same, but the factorization integral changes the shape of $d \sigma(Z) /d Q^2$.

Cheers,
T.

9. Luboš Motl - November 26, 2008

Dear Tommaso,

that sounds somewhat contrived to me. How could have we known that the y-axis captures a quantity as unnatural as “average Z mass produced in pp collisions”? Indeed, this quantity may be biased and rapidity-dependent. (It is an even function because of the symmetry between the two beams.)

I was assuming that you use the best information you have to reconstruct the Breit-Wigner from your measurements, and the y-axis of the last graph was the average mass in the Breit-Wigner distribution which should be rapidity-independent whenever you have enough statistics.

Do you personally assume/think that the Y=0 value of the “average produced Z mass” from your last graph should be the same thing as the central value of the Breit-Wigner distribution? I don’t quite see the reason. For example, if you collided two gluons whose CMS energy would be systematically higher than the Z mass and only sometimes dropped close to the Z mass, you would usually produce Z resonances that would be above the Z mass Breit-Wigner central value, wouldn’t you? And vice versa.

Best
Lubos

10. dorigo - November 26, 2008

Hi Lubos,

the quantities in the graphs have nothing to do with measurement: they are just the result of a simple integration of the product of parton densities and Breit-Wigner. The result of the integration does depends on rapidity.

As for your last question, no, of course I do not assume that the Y=0 value of the average mass is 91.186 GeV. In fact, the whole point of this stupid study is to show the downward offset of the whole average, and capture its dependence on Y, to use that information in our calibration studies.

As for two gluons producing the Z… Well. That does happen, but not at leading order in alpha_s. You need to emit an extra gluon.

Cheers,
T.

11. Luboš Motl - November 26, 2008

OK, thanks, makes sense, but again, it is my feeling that I would not be the only one who thinks that the Y=0 value of your last graph is your believed/assumed/deduced/measured (according to the origin of the data) Breit-Wigner central value. You may organize a poll.

Yes, I know that the simplest diagram with two gluons to Z production has a loop.

12. dorigo - November 26, 2008

Nope, Lubos, the simplest diagram with two gluons to Z production does not just have a loop, it also has an outgoing gluon flanking the Z.

Cheers,
T.

13. carlbrannen - November 26, 2008

I’m a little confused by the post in that it says that this is for hadron collisions but doesn’t say what hadrons. I would have thought p and p-bar, but then you’d see u-ubar valence collisions. So is this a p-p collider? Maybe I didn’t read it as carefully as needed.

14. Luboš Motl - November 26, 2008

Dear Tommaso, I didn’t say that it didn’t have an outgoing gluon flanking the Z besides the loop (or other extra particles).

In fact, I am saying the opposite (now). Perform a Z2 symmetry W changing all the electroweak charges to minus themselves, which exists at least at the tree and one-loop level, to see that 1 Z boson is W-odd while two gluons are W-even. 😉

At any rate, whatever is the right clarification here, don’t feel the paranoia. I am not disagreeing with you, seeing it’s impossible to draw non-vanishing 1-loop diagrams with the 3 particles only.

Carl, except for a few initial paragraphs, Tommaso’s text is about the CMS detector which is a detector at the LHC which is a collider at CERN which is a facility on the Franco-Swiss border. And the LHC collides two protons, indeed. That’s a difference from the Tevatron which collided p and p-bar. 😉

15. dorigo - November 26, 2008

Lubos, try again… I am happy you are not disagreeing with me, but when you say “seeing it’s impossible to draw non-vanishing 1-loop diagrams with the 3 particles only.” makes ME disagree with you.
There of course are 3-particle 1-loop diagrams which give non-negligible contributions to cross sections, in some cases.

Cheers,
T.

16. dorigo - November 26, 2008

Hi Carl,

no it’s not your fault… Indeed I did not specify well enough. In any case, u-ubar collisions happen with any hadron, even glueballs would contain some of those quarks, in their own sea.

Cheers,
T.

17. Luboš Motl - November 27, 2008

Tommaso, I guess that you know what I meant, you agree, and you’re just teasing me.

18. dorigo - November 27, 2008

Hi Lubos,

of course I am teasing you! After proving you are fallible, it has become a quite enjoyable occupation.

Your sentence “it’s impossible to draw non-vanishing 1-loop diagrams with the 3 particles only.” is wrong, and I just pointed it out. Otherwise the main production process for the Higgs boson at hadron colliders would be non-existent!

Cheers,
T.

19. Luboš Motl - November 27, 2008

Dear Tommaso,

human beings are fallible almost by definition. In some cases it may take up to 35 years or so to demonstrate this fact, but it is possible.

Still, I think that it would be wiser from you to work on the problematic features of your CDF paper and similar work. The particular bugs have increased by a factor of three by your proof of my fallibility. The cross section of the ghost events is 0 pb and not 70 or even 200 pb.

To do so, it is indeed a wise idea to be accurate about various sentences – but you should require the accuracy not only from me but also from yourself and your collaborators. This very posting of yours was enough to obfuscate what collider you were talking about and what was the y-axis of one graph that you found important enough to include but irrelevant enough not to explain.

Not sure whether you have already had time to look at my questions about your definition of “the” 2 muons in 3-muon events and related problems that it brings, besides the tables in the paper that are later shown incorrect but that are still treated as correct sources of numbers.

Cheers
LM

20. dorigo - November 28, 2008

Hi Lubos,

it took me zero time to prove you fallible, since the xs issue was the first time we disagreed on something that lent itself to proof.

I had a very full week, but this weekend I will put together another post on the ghost muons, as well as an attempt at answering your questions, which do deserve it. It was not my intention to make you “expiate” your stubbornness with the xs issue by having you wait, but things worked that way.

Your point about the general cryptic nature of experimental plots and the need for more exhaustive explanations is well taken.

Cheers,
T.

21. pedro - November 28, 2008

Tommaso and Lubos,

I agree with the point about the cryptic nature of experimental plots. However, it is very easy to be cryptic. We can do that by not showing all the details but also by showing too many details. So, one way out is for interested readers to ask question. I am sure that inside the collaboration lots of questions arises and the papers is clarified for those readers, the final draft is much better. But the improvement are to satisfy the inside readers.

Cheers,
Pedro.
PS. Tommaso, thaks for the answer respected to the x distribution. Now I understand (and understood also that this information was hidden, for my own benefit, from me).

22. Ramat - November 30, 2008

Ok but remove the grey background in ROOT default setting..
gROOT->SetStyle(“Plain”);
😉

23. dorigo - December 18, 2008

Ramat, thanks for the advice, will do…
Cheers,
T.

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